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AP/AD >= 1 - BC^2/(AB^2 + CA^2) , altitudes related

Source: Indonesia INAMO Shortlist 2009 G6 https://artofproblemsolving.com/community/c1101409_

December 10, 2021
geometrygeometric inequalityaltitudes

Problem Statement

Suppose the points D,E,FD, E, F lie on sides BC,CA,ABBC, CA, AB, respectively, so that AD,BE,CFAD, BE, CF are the altitudes. Also, let ADAD and EFEF intersect at PP. Prove that APAD≥1−BC2AB2+CA2\frac{AP}{AD} \ge 1 - \frac{BC^2}{AB^2 + CA^2}
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